In the Downing Street garden, outside David Cameron’s office window, is a giant red light bulb in the middle of the lawn.

His children use the “modern art installation” as an impromptu goal, but it also serves as a reminder of the constant need for the Prime Minister to keep thinking of radical solutions to the unprecedented economic and political challenges he faces.

Many in his own party are increasingly of the opinion that Downing Street and the Government’s key figures may be failing to rise to the challenge.

Britain is back in recession and the crisis in the euro looks likely to condemn this country to a lost economic decade. Mr Cameron does not “see a time” when the austerity programme will end.

The budget was regarded as a presentational disaster and George Osborne, the Prime Minister’s key lieutenant and ally, is facing questions over his position that seemed unthinkable last year.

Mr Cameron is struggling to persuade even his own MPs to support government policy and there is a growing clamour for a more Right-wing agenda.

The relationship with the Liberal Democrats appears to be in a state of permanent limbo with relaunches for the Coalition an almost monthly occurrence. After a frenetic first 18 months bringing radical new policies for welfare, schools and health, Downing Street aides talk of “mid-term blues” as Labour remain consistently ten points ahead in the polls.

TORY DISQUIET WITH THE LIB DEMS

Mr Cameron says he understands the grumbling and that even he becomes “frustrated” that he cannot always get his own way. But he urges his supporters and the public to give him time to see the fruits of the Coalition’s work.

“If you’re in government, you don’t get everything you want,” he says. “I’m the Prime Minister; I don’t get everything I want. Sometimes you have to, we have to, hammer out compromises around the Cabinet table, or in your department. And, if you lose the argument, you have to support the position.

“Obviously it’s been a very tough time to be in government but I think the most important thing is we’ve given Britain a secure plan for getting out of the debt and the deficit and the difficulties that we inherited.

“The sense I get of the mood of the British public is of very fair-minded people: they say it’s very difficult, they’re quite pessimistic about the state of the world and the state of the world economy and their own finances. They know it’s tough, they know it’s difficult, they know that none of these things are going to be easily done.”

Mr Cameron appears to have some sympathy for the dozens of Conservative MPs who defied the whips to vote against policy on Lords reform and Europe.

Although the rebels are highly unlikely to be offered government jobs in the reshuffle expected this autumn, the Prime Minister indicates that their careers have not been forever blighted.

“If no one who ever voted against the Government [got a] ministerial office ... [Iain] Duncan-Smith wouldn’t be in the Cabinet. I think politics has changed, there are people who come with more strong and independent views. It’s difficult in coalition because sometimes you’re coming forward with proposals that your coalition partners are more enthusiastic about than you are.” The comments are likely to intensify Lib Dem suspicions that the Prime Minister is not really intent on a hard-line approach to ensure that the Coalition’s policy deals become reality.

THE FUTURE OF THE COALITION

Mr Cameron has obviously been dedicating a lot of time to thinking about the changing nature, and possibly form, of the Coalition over the next three years. Key decisions on airports, wind farm subsidies and social care have been delayed in recent weeks, and future assumptions about how the relationship will function are being reconsidered.

For example, the Coalition was due to agree a programme for public spending from 2014 to 2017. Discussions are due to begin later this year or early next year, but the prospect of a wide-ranging, three-year deal now seems remote. Instead, the Prime Minister suggests a shorter agreement may be under consideration.

“We talked about the need to have one. But how many years it runs for, how exactly it works, we’re going to spend more time talking about that. I think what matters is taking the big decisions about the future. I much prefer to approach it from the decisions, and then work out how we are going to [do it], how many years. I want us to make some very big decisions about housing policy, welfare policy, how we fund infrastructure. Let’s work out how these decisions trade off against each other.”

He also indicates that the Coalition may function differently in the year before an election. His preference is for the current arrangement to run to May 2015. But asked whether it may switch to “confidence and supply”, where Lib Dems would not vote to bring the Government down but would not support a policy agenda, he replies: “I think the Coalition will run to 2015. How exactly you separate before an election and fight an election: to be determined.”

THE ECONOMY

Away from Westminster, the fate of the Cameron government is largely at the mercy of international events.

On Tuesday morning, shortly before flying to Afghanistan, the Prime Minister had a “long meeting about the eurozone crisis and how that’s going to develop over the summer and autumn”.

The prognosis is not good. There are three scenarios: the first is the eurozone acting together to sort out the problems, the third is the break-up of the single currency. The British assessment is that economic malaise is likely. The Prime Minister says: “I think scenario two, which we are in at the moment … steps taken to make do and mend, steps to kick the can down the road, steps that bring about a bit of banking union or fiscal union but the crisis continues. I think that is the most likely scenario.

“And that’s bad for Britain, it’s bad for Europe. But we have to recognise that what these countries are engaged in is an incredibly difficult decision. This is why we didn’t join the euro.

“What is required economically is extremely tough to deliver politically because effectively the countries of the South are asking Germany to take on some of their debts and responsibilities. And, the Germans are asking the countries of the South to give up their economic sovereignty.”

The turmoil means the Prime Minister is increasingly resigned to another five-year austerity programme after the next election. Previously, the Chancellor had pledged that spending cuts would be complete by 2015. Last year, he extended that to 2017 amid a deteriorating economic situation. Will this now be a ten-year austerity programme?“This is a period for all countries, not just in Europe but I think you will see it in America too, where we have to deal with our deficits and we have to have sustainable debts. I can’t see any time soon when the pressure will be off. I don’t see a time when difficult spending choices are going to go away.”

EUROPE

A growing number of Conservatives, including many MPs, blame many of the problems facing the British economy on unnecessary European interference and regulation. They are demanding an in-or-out referendum which would pave the way for Britain to leave the EU.

The Prime Minister insists that those calling for an immediate referendum have a “perfectly honourable position”. He wants to negotiate a “new settlement” with the EU with powers returned to Britain. But Mr Cameron will not countenance leaving the EU and says he would never campaign for an “out” vote in a referendum.

“I think it would be bad for Britain,” he says. “When I look at what is in our national interest, we are not some country that looks in on ourself or retreats from the world. Britain’s interest – trading a vast share of our GDP – is to be in those markets. Not just buying, selling, investing, receiving investment but also helping to write the rules. If we were outside, we wouldn’t be able to do that.”

He adds: “It comes back to this, who are going to be the winning nations for the 21st century? If your vision of Britain was that we should just withdraw and become a sort of greater Switzerland, I think that would be a complete denial of our national interests.”

The OLYMPICS

Mr Cameron has claimed that he will use public transport during the Olympics rather than the controversial Games lanes. He said he was driving his security team “mad” by abandoning his official car in favour of the Underground.

There are growing fears that special lanes reserved for accredited Olympics vehicles will cause gridlock for normal motorists in the capital. Asked whether he would be using the so-called “Zil lanes”, named after the limousines used by Soviet leaders, the Prime Minister said: “We are banished from using that terminology. I went in a Games lane, but it was not acting as a Games lane when I came back from Chequers on Sunday. We were happily driving on it because everyone else was driving on it.

“I will be using public transport and that is how I recommend all Telegraph readers travel, use the javelin trains. I now use the Tube, I drive my protection team mad in London by saying why on earth are we in the car, the Tube is quicker.”

Mr Cameron will spend much of his time during the Olympics meeting business leaders from around the world who will be in London.

They include Eric Schmidt, the chairman of Google; John Chambers, the chief executive of Cisco Systems; Vikram Pandit, the chief executive of Citigroup; and Satoshi Ozawa, the chief executive of Toyota. Hundreds of millions of pounds worth of new business deals are expected to be agreed.